Operational Quantum Theory II - Relativistic Structures

by: Heinrich Saller

Springer-Verlag, 2006

ISBN: 9780387346441 , 333 Pages

2. Edition

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Operational Quantum Theory II - Relativistic Structures


 

Contents

7

INTRODUCTION

13

MATHEMATICAL TOOLS

24

1 LORENTZ OPERATIONS

28

1.1 Spacetime Lie Algebras

29

1.2 Left- and Right-Handed Weyl Spinors

31

1.3 Finite-Dimensional Representations of the Lorentz Operations

33

1.4 Spacetime Translations as Spinor Transformations

37

1.5 Minkowski Cli.ord Algebras

42

1.6 Dirac Spinors and Dirac Algebra

43

1.7 Re.ections for Position and Time

45

1.8 Dirac Equation

50

1.9 Polynomials with Lorentz Group Action

51

1.10 Summary

54

1.11 Doubled Lie Algebra

56

1.12 Conjugate-Adjoint Representations

57

2 SPACETIME AS UNITARY OPERATION CLASSES

58

2.1 Spacetime Translations

58

2.2 Nonlinear Spacetime

62

2.3 Spacetime and Hyperisospin

64

2.4 Orbits and Fixgroups of Hyperisospin

67

2.5 Orbits and Fixgroups in Spacetime

71

2.6 Summary

78

2.7 Fixgroups of Representations

79

2.8 Orbits with Signatures

79

2.9 Fix- and Stabil-Lie Algebras

80

2.10 Transmutators as Coset Representations

81

3 PROPAGATORS

84

3.1 Point Measures for Energies

84

3.2 Relativistically Distributed Time Representations

86

3.3 Fourier Transforms of Energy- Momentum Distributions

87

3.4 Scattering Waves (on Shell)

90

3.5 Macdonald, Neumann, and Bessel Functions

91

3.6 Yukawa Potential and Force (o. Shell)

93

3.7 Feynman Propagators

95

3.8 Summary

96

3.9 Distributions

97

3.10 Fourier Transformation

100

3.11 Measures of Symmetric Spaces

101

Bibliography

103

4 MASSIVE PARTICLE QUANTUM FIELDS

105

4.1 Quantum Bose and Fermi Oscillators

107

4.2 Relativistic Distribution of Time Representations

111

4.3 Quantum Fields for Massive Particles

112

4.4 Lorentz Group Embedding of Spin

116

4.5 Massive Spin-

119

Particle Fields

119

4.6 Massive Spin-1 Particle Fields

122

4.7 Massive Spin-1/2 Dirac Particle Fields

124

4.8 Massive Spin-1/2 Majorana Particle Fields

127

4.9 Spacetime Re.ections of Spinor Fields

128

4.10 Representation Currents

129

4.11 Relativistic Scattering

134

4.12 Summary

138

Bibliography

140

5 MASSLESS QUANTUM FIELDS

141

5.1 Noncompact Time Representations in Quantum Algebras

142

5.2 Inde.nite Metric in Quantum Algebras

146

5.3 Relativistic Distributions of Noncompact Time Representations

148

5.4 The Hilbert Spaces for Massless Particles

150

5.5 Massless Scalar Bose Particle Fields

151

5.6 Massless Scalar Fermi Fields ( Fadeev- Popov Fields)

153

5.7 Polarization (Helicity) in Spacetime

154

5.8 Massless Weyl Particle Fields

156

5.9 Massless Vector Bose Fields ( Gauge Fields)

157

5.10 Eigenvectors and Nilvectors in a Gauge Dynamics

162

5.11 Summary

166

Bibliography

166

6 GAUGE INTERACTIONS

167

6.1 Classical Maxwell Equations

168

6.2 The Electromagnetic Gauge Field

171

6.3 The Charged Relativistic Mass Point

175

6.4 Electrodynamics as U(1)-Representation

176

6.5 Quantum Gauge Fields

180

6.6 Representation Currents

180

6.7 Lie-Algebra-Valued Gauge Fields

181

6.8 Lie Algebras of Spacetime and Gauge Group

184

6.9 Electroweak and Strong Gauge Interactions

186

6.10 Ground State Degeneracy

188

6.11 From Interactions to Particles

192

6.12 Reflections in the Standard Model

200

6.13 Summary

203

6.14 Fadeev- Popov Degrees of Freedom

204

6.15 Gauge and BRS-Vertices

208

6.16 Cartan Tori

210

Bibliography

214

7 HARMONIC ANALYSIS

216

7.1 Representations on Group Functions

219

7.2 Harmonic Analysis of Finite Groups

222

7.3 Algebras and Vector Spaces for Locally Compact Groups

225

7.4 Harmonic Analysis of Compact Groups

227

7.5 Hilbert Representations and Scalar- Product- Inducing Functions

231

7.6 Harmonic Analysis of NoncompactGroups

235

7.7 Induced Group Representations

238

7.8 Harmonic Analysis of Symmetric Spaces

247

7.9 Induced Representations of Compact Groups

250

7.10 Representations of A.ne Groups

254

7.11 Group Representations on Homogeneous Functions

264

7.12 Harmonic Analysis of Hyperboloids

269

7.13 Convolutions

273

7.14 Abelian Convolution of Functions and Distributions

274

7.15 Parabolic Subgroups

276

Bibliography

277

8 RESIDUAL SPACETIME REPRESENTATIONS

279

8.1 Linear and Nonlinear Spacetime

280

8.2 Residual Representations

282

8.3 Residual Representations of the Reals

290

8.4 Residual Representations of Tangent Groups

292

8.5 Residual Representations of Position

294

8.6 Residual Representations of Causal Spacetime

298

8.7 Time and Position Subgroup Representations

302

Bibliography

305

9 SPECTRUM OF SPACETIME

307

9.1 Convolutions for Abelian Groups

308

9.2 Convolutions for Position Representations

310

9.3 Convolution of Singularity Hyperboloids

313

9.4 Convolutions for Spacetime

315

9.5 Tangent Structures for Spacetime

322

9.6 Translation Invariants as Particle Masses

329

9.7 Normalization of Translation Representations

333

MATHEMATICAL TOOLS 9.8 Divergences in Feynman Integrals

336

Bibliography

338

Index

339